Martin (1992: 498):
Register is defined as "the configuration of semantic resources that the member of a culture associates with a situation type. It is the meaning potential that is accessible in a given social context" (Halliday 1978: 111).
Defining register in these terms pushes considerations of context such as those addressed by Malinowski and Firth one level up, to what Halliday refers to as context of situation — presumably what is referred to as situation (non-linguistic phenomena) in Fig. 7.4. Context of situation is then organised metafunctionally into field, tenor and mode as described above.
This is a very strange misunderstanding indeed. Halliday's definition of register — as the meaning that realises a situation type — has no bearing whatsoever on the organisation of strata. Register and situation type are midway points on the cline of instantiation — for the strata of semantics and context, respectively.
Halliday's notion of context derives from Malinowski and Firth, and is structured semiotically in terms of field, tenor and mode, as Martin has already explained (p494) with a quote from Halliday (1978).